56
3 Mechanical Aspects of Biosystems
Surface forces act only through the faces of the rectangular elements and come from
interaction with adjacent elements.
The surface forces acting across each surface of a given small mass element
will be summed into a single force acting on that surface. In the spirit of the
approximation of small mass elements, the surface forces will not vary significantly
across a single element’s surface. This means that if any element’s surface were cut
in half, the force on that half surface should be half the force on the whole. Thus,
the surface force per unit area should not be affected by the choice of size for the
mass elements, as long as they are small enough. Similarly, the body force per unit
volume should not depend on the choice of size of the mass element, as long as they
are small enough. It is for these reasons that Newton’s second law for the behavior
of a distortable material is given a form that is independent of the choice of small
mass elements.
Instead of writing Newton’s 2nd law for the masses, we write a corresponding
expression by dividing each side of Newton’s 2nd law for a small mass element
by its volume. The law is then expressed in terms of only ‘intensive variables’ 31 :
body force per unit volume; the change across the material of surface forces per
unit area; and the density of the material. We should expect that the laws of nature
do not depend on sizes which we are able to pick arbitrarily, such as the size of
the mass elements inside a body. Our choice of small mass elements is arbitrary, as
long as their size is small enough that quantities which change across the body do
not change significantly across the small mass elements, and large enough to make
averages over atoms meaningful.
In Fig. 3.8, a pair of the surface forces denoted by F z are shown acting on the
surface of an approximate cubic element of material, with the two surfaces on which
this pair of forces act lying perpendicular to the z-axis of the selected coordinate
system. The unit vectors n are taken normal to the surface. Notice that F z may have
components along all three axes. The stress (force per area) on the ‘top’ surface in
the figure is then S z = F z /(dxdy) = F 3 /A 3 , where A 3 = dx dy by using (1, 2, 3)
as indices for (x, y, z), and k can replace (i, j ) as the ‘missing’ third index in
(i, j, k). The components of this stress will be named S 3 ≡ {S 31 , S 32 , S 33 }; similarly
for S 1 and S 2 . Thus,
S ij ≡ F ij /A i ,
(3.15)
where F ij is the j th component of the force F i , and A i is the area element
perpendicular to the i-th direction. Notice that the ‘diagonal’ components of the
stress are pressures on each side of the cube: S 11 = S 22 = S 33 = −p.
By adding the forces on all six sides of the distorted cube, one finds that the net
force will be
δF = −∇ · S δV ,
(3.16)
31 Intensive variables do not depend on the size of selected elements; ‘extensive variables’ do.
3 Mechanical Aspects of Biosystems
Surface forces act only through the faces of the rectangular elements and come from
interaction with adjacent elements.
The surface forces acting across each surface of a given small mass element
will be summed into a single force acting on that surface. In the spirit of the
approximation of small mass elements, the surface forces will not vary significantly
across a single element’s surface. This means that if any element’s surface were cut
in half, the force on that half surface should be half the force on the whole. Thus,
the surface force per unit area should not be affected by the choice of size for the
mass elements, as long as they are small enough. Similarly, the body force per unit
volume should not depend on the choice of size of the mass element, as long as they
are small enough. It is for these reasons that Newton’s second law for the behavior
of a distortable material is given a form that is independent of the choice of small
mass elements.
Instead of writing Newton’s 2nd law for the masses, we write a corresponding
expression by dividing each side of Newton’s 2nd law for a small mass element
by its volume. The law is then expressed in terms of only ‘intensive variables’ 31 :
body force per unit volume; the change across the material of surface forces per
unit area; and the density of the material. We should expect that the laws of nature
do not depend on sizes which we are able to pick arbitrarily, such as the size of
the mass elements inside a body. Our choice of small mass elements is arbitrary, as
long as their size is small enough that quantities which change across the body do
not change significantly across the small mass elements, and large enough to make
averages over atoms meaningful.
In Fig. 3.8, a pair of the surface forces denoted by F z are shown acting on the
surface of an approximate cubic element of material, with the two surfaces on which
this pair of forces act lying perpendicular to the z-axis of the selected coordinate
system. The unit vectors n are taken normal to the surface. Notice that F z may have
components along all three axes. The stress (force per area) on the ‘top’ surface in
the figure is then S z = F z /(dxdy) = F 3 /A 3 , where A 3 = dx dy by using (1, 2, 3)
as indices for (x, y, z), and k can replace (i, j ) as the ‘missing’ third index in
(i, j, k). The components of this stress will be named S 3 ≡ {S 31 , S 32 , S 33 }; similarly
for S 1 and S 2 . Thus,
S ij ≡ F ij /A i ,
(3.15)
where F ij is the j th component of the force F i , and A i is the area element
perpendicular to the i-th direction. Notice that the ‘diagonal’ components of the
stress are pressures on each side of the cube: S 11 = S 22 = S 33 = −p.
By adding the forces on all six sides of the distorted cube, one finds that the net
force will be
δF = −∇ · S δV ,
(3.16)
31 Intensive variables do not depend on the size of selected elements; ‘extensive variables’ do.